Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
851 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4⋊C8 | C2 | semidirect |
| 5. #1851 | C2 | direct (N × H) |
| 6. C2×#336 | C2 | semidirect |
| 17. C2×C4⋊C8 | C22 | semidirect |
| 24. #336 | C22 | semidirect |
| 80. C22×C4 | M16 | semidirect |
| 142. C2×C4 | C2×M16 | semidirect |
| 174. C2 | #1851 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×C4⋊C8 | 2 |
| C22×C22⋊C8 | 2 |
| C2×C4×C22⋊C4 | 2 |
| #1851 | 2 |
| C2×#336 | 2 |